Lesson 3 7
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Lesson 3-7. Higher Order Deriviatives. Objectives. Find second and higher order derivatives using all previously learned rules for differentiation. Vocabulary. Higher order Derivative – taking the derivative of a function a second or more times. Example 1.

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Lesson 3-7

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Lesson 3 7

Lesson 3-7

Higher Order Deriviatives


Objectives

Objectives

  • Find second and higher order derivatives using all previously learned rules for differentiation


Vocabulary

Vocabulary

  • Higher order Derivative – taking the derivative of a function a second or more times


Example 1

Example 1

Find second derivatives of the following:

  • y = 5x³ + 4x² + 6x + 3

  • y = x² + 1

y’(x) = 15x² + 8x + 6

y’’(x) = 30x + 8

y’(x) = ½ (x² + 1)-½ (2x) = x / (x² + 1)½

(1)(x² + 1)½ - x ½ (x² + 1) -½ (2x)

y’’(x) = --------------------------------------------

(x² + 1)


Example 2

Example 2

Find the fourth derivatives of f(x) = (1/90)x10 + (1/60)x5

Find a formula for f n(x) where f(x) = x-2

f’(x) = (10/90)x9 + (5/60)x4

f’’(x) = (90/90)x8 + (20/60)x3

f’’’(x) = (720/90)x7 + (60/60)x2

F’’’’(x) = (5040/90)x6 + (120/60)x = 56x6 + 2x

f’(x) = (-2)x-3

f’’(x) = (-2)(-3)x-4

f’’’(x) = (-2)(-3)(-4)x-5

f’’’’(x) = (-2)(-3)(-4)(-5)x-6

fn(x) = (-1)n(n+1)! x-(2-n)


Example 3

Example 3

Find the third derivatives of the following:

y = (3x³ - 4x² + 7x - 9)

f(x) = e2x+7

y’(x) = 9x² - 8x + 7

y’’(x) = 18x - 8

y’’’(x) = 18

f’(x) = 2e2x+7

f’’(x) = 4e2x+7

f’’’(x) = 8e2x+7


Chain rule revisited

Chain Rule Revisited

FunctionDerivativeFunctionDerivative

y = xny’ = n xn-1y = uny’ = n un-1 u’

y = ex y’ = exy = euy’ = u’ eu

y = sin xy’ = cos xy = sin uy’ = u’ cos u

y = cos xy’ = -sin xy = cos uy’ = -u’ sin u

y = tan xy’ = sec² xy = tan uy’ = u’ sec² u

y = cot xy’ = -csc² x y = cot uy’ = u’ csc² u

where u is a function of x (other than just u=x) and u’ is its derivative


Derivatives faq

Derivatives FAQ

  • When do I stop using the chain rule?Answer: when you get something that you can take the derivative of without having to invoke the chain rule an additional time (like a polynomial function).

  • Does the argument in a trig function ever get changed?No. The item (argument) inside the trig function never changes while taking the derivative.

  • Does the exponent always get reduced by one when we take the derivative?Only if the exponent is a constant! If it is a function of x, then it will remain unreduced and you have to use another rule instead of simple power rule.

  • When do I use the product and quotient rules?Anytime you have a function that has pieces that are functions of x in the forms of a product or quotient.


Summary homework

Summary & Homework

  • Summary:

    • Derivative of Derivatives

    • Use all known rules to find higher order derivatives

  • Homework:

    • pg 240 - 242: 5, 9, 17, 18, 25, 29, 49, 57


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